Representation theory of two families of quantum projective 3-spaces

Stephenson and Vancliff recently introduced two families of quantum projective 3-spaces (quadratic and Artin–Schelter regular algebras of global dimension 4) which have the property that the associated automorphism of the scheme of point modules is finite order, and yet the algebra is not finite over its center. This is in stark contrast to theorems of Artin, Tate, and Van den Bergh in global dimension 3. We analyze the representation theory of these algebras. We classify all of the finite-dimensional simple modules and describe some zero-dimensional elements of Proj, i.e., so called fat point modules. In particular, we observe that the shift functor on zero-dimensional elements of Proj, which is closely related to the above automorphism, actually has infinite order.